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Fugue No. 10E minorWell-Tempered Clavier Book IJohann Sebastian Bach

2002 Timothy A. Smith (the author)1

To read this essay in its hypermedia format, go to the Shockwave movie athttp://www2.nau.edu/tas3/wtc/i10.html .

Subject: Fugue No. 10, Well-Tempered Clavier, Book I

This is the only two-voice fugue in the WTC. Because a fugue requires atleast two voices for its subject and answer relationship, the voicing isminimalistic. When it comes to counterpoint however this is one of Bach's mostclever of fugues. In this analysis I'll demonstrate how it is:

two fugues in one

double counterpointed at the octave a canon like a Mbius strip desirous of a coda

Two Fugues in OneThe timeline reveals this fugue to be highly symmetrical: four sections in an

abab pattern. All of the music is encapsulated in the first nineteen measures.These contain an exposition and 1st development (line 1) followed by a re-exposition and 2nd development (line 2). Thus the first two lines comprise what Ishall call the fugue.

What remains is double counterpoint of the fugue transposed to new keys.Containing no new music, line 3 and line 4 are derived from the fugue by doublecounterpoint. For this reason I have identified them as the counterfugue. So youcan see that this fugue is really two fugues in one.

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so long as credit is given to the author as per fair use. You may not reproduce this documentelectronically, enfold it into a web site, or incorporate it into a saleable product without writtenpermission from the author.

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Double Counterpointed at the OctaveDouble counterpoint involves an exchange of registers; the melody that had

been in the high voice is now heard in the low and vice versa. If it is helpful, youmay think of double counterpoint like a game of leapfrog. At the beginning youleapt over the person in front of you. At the moment of your leap you were higher

than him and he lower than you. Now it has come his turn to leap and thesituation is reversed. This is what I meant by an exchange of registers.The counterfugue is comprised entirely of double counterpoint at the octave.

Bach has simply repeated the first nineteen measures with the high voice havingmoved to the low and vice versa.

In addition to its double counterpoint the counterfugue has also undergonetransposition. Line 3 is a transposition of line 1 up a 4th. By contrast line 4 is atransposition of line 2 in the opposite direction, down a 4th.

If double counterpoint were new to you it would be instructive to compareanalogous measures of the fugue and counterfugue. You can do this by holdingthe mouse down and hovering over corresponding measures in the timeline.

To find the corresponding measure in the counterfugue, add 19 to eachmeasure in the fugue. So m. 11 of the fugue corresponds with m. 30 (11+19=30)of the counterfugue. Corresponding measures are vertically aligned andseparated by one line, the even-numbered lines corresponding with each other,as do the odd.

A CanonIt is in the nature of a fugue to be imitative. But this fugue is more so than

most. This is because it is also a canon. So that you may understand thisrelationship I should like to clarify the difference between a canon and a fugue.

The relationship between a fugue's subject and its answer is essentially oneof imitation. In a tonal answerthe contour is imitated with occasional intervalmutation in order to maintain the same key. In a real answerthe intervals andcontour are identical, but transposed to a new key. In most fugues such strictimitation normally breaks down after the exposition.

Like the fugue, a canon generates one voice out of another by contrapuntalmeans. Whatever means is chosen (a different interval, rhythmic proportion,contrary motion, retrograde), the second voice is bound as if by law to continueimitating the first voice by the same means. This is what "canon" means: rule orlaw. The moment it breaks the rule, the episode is no longer canonic.

Remember this. A composition can be: (1) a canon and not a fugue, (2) afugue and not a canon, or (3) both a canon and a fugue. Compositions in thethird category are rare, therefore of special importance. This fugue belongs tothat category; it is at the same time a canon and a fugue.

As a fugue this work employs a realrelationship. The answer (low voice mm.3-4) is a transposition of the subject (high voice mm. 1-2) from the key of em tobm. This little segment is a canonic episode; it does not qualify the entire workas a canon because the imitation is abandoned in m. 7.

Here is what makes this fugue a canon. Earlier I said that all of the music iscontained in mm. 1-19. But there is another way to hear it. All of the music is

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also contained in the high voice. Let us call lines 1-2 of the high voice Part A,and lines 3-4 part B. Everything to the Coda could be represented as follows:

Part A Part B (high voice)Part B Part A (low voice)

So what I had called the fugue is nothing less than column one (above) andthe counterfugue is column two. The columns illustrate how double counterpointundergirds the canonic process; it is impossible to have a canon without it. Sothis fugue is a canon because it consists entirely of double counterpoint (Codaexcluded).

Many of the canonic structures in this fugue have been animated. One voicegravitating to another indicates canon. Motions that are limited to the same linesindicate small-scale imitation (canonic episodes). Motions involving whole linesmigrating to other lines indicate large-scale imitations as in: m. 1, m. 11, m. 20and m. 30. These larger imitations are what make this fugue a canon.

Like a Mbius StripNow let's talk about lines. If you draw a line from Eisenach

(Bach's birthplace) to Leipzig (where he died), you will comeacross Schulpforta, one of those delightful little Saxon townswith which Bach would have been very familiar. It was inSchulpforta that the mathematician August Mbius was born in1790. Interestingly Mbius taught at Leipzig University whereBach had directed the Collegium Musicum 100 years before.Both men died in Liepzig.

While in Leipzig, Mbius answered a question on the geometric theory ofpolyhedra posed by the Paris Academy. His answer required the calculation ofproperties left unchanged by the continuous deformation of lines in space. Hisproof referenced what has since come to be known as the Mbius strip.

A Mbius strip is a surface that has only one side, like a sheet of paper with afront but no back. Here's how to make one. Cut a narrow slip of paper then writeFUGUEon one surface and COUNTERFUGUEon the other. Tape the endstogether with a 180-degree twist (as in the diagram bottom right).

Now draw a line paralleling the length of the strip. Take care not to lift yourpencil from the surface. Keep drawing until the line has returned to its point oforigin. Amazingly the line will have marked both "sides" of the strip. That youcould do this without lifting your pencil proves that the Mbius strip really has oneside and not two.

In the course of drawing the line you will have traversed through both theFUGUE andCOUNTERFUGUE. The Mbius strip illustrates how these two arereally made of the same music. It also illustrates the seamless manner in whichBach has moved from one to the other.

Measure 39 is very important to our understanding of this fugue. If youcompare it with m. 1 you will hear that it is the point of return. It is like that pointin the Mbius strip where the line returns to its point of origin. Continuation of the

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process beyond m. 39 would have required another traverse of Bach's fugalMbius strip. Thankfully he has interrupted the process and brought the fugue toa close by means of a four-measure coda.

The Mbius strip accomplishes its magic by means of its twist. The twist iswhat gives its plane a single side. Bach's fugue also has a twist that allows it to

move seamlessly from fugue to counterfugue and back.To appreciate the significance of Bach's twist it is important to emphasize thatthe fugue and counterfugue are analogous to each other in every way except forone interval. That interval is found in beat one of m. 10: a rising third betweenthe 2nd and 3rd pitches. At the analogous point in m. 29 Bach has composed arising fourth. It is a tiny difference but one that allows the counterfugue to returnto the fugue in the correct key (m. 39). This return is abundantly clear in acomparison of the following pairs of measures.

m. 15 - m. 35m. 16 - m. 36

m. 17 - m. 37

Desirous of a CodaThe significance of the interval deviation in m. 10 and m. 29 cannot be

overstated. I call it, "the twist." You may have observed that the twist in theMbius strip (bottom right) occurs in these measures.

The fugue leads naturally to the counterfugue. But without the twist, thecounterfugue would not have returned to the fugue in its original key. Without thetwist this work would have been a modulatingorspiralcanon. It would eventuallyhave gotten to the right key but only after several more repetitions.

The twist turns this work instead into what is known as aperpetualcanon--aclosed and recursive system like that of the Mbius strip. Like the Mbius stripthis fugue has a proclivity to repeat itself forever. Such a tendency is illustratedby the following comparison (DC stands forDouble Counterpoint):

L1 is the DC of L3 down a 4thL2 is the DC of L4 up a 4thL3 is the DC of L1 up a 4thL4 is the DC of L2 down a 4th

How is it all to end? I jest that the fugue is desirous of a coda. I rather thinkthat the fugue would be quite content to repeat itself forever. In truth it is we whodesire it to end. So by the insertion of a Coda Bach has brought it to a satisfying,and welcome, close.